Asymptotically equicontinuous sequences of operators and a Banach-Steinhaus type theorem

2023/02/13 by Mashreghi, Javad, Ransford, Thomas
#40H05 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2302.06722

Abstract

We introduce the notion of an asymptotically equicontinuous sequence of linear operators, and use it to prove the following result. If X,Y are topological vector spaces, if T_n,T:X→ Y are continuous linear maps, and if D is a dense subset of X, then the following statements are equivalent: (i) T_nx→ Tx for all x∈ X, and (ii) T_n x→ Tx for all x∈ D and the sequence (T_n) is asymptotically equicontinuous.

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